Consider some $f: [0,1)\times [0,1)\to \mathbb{R}$. I'm interested in conditions that guarantee that the following iterated one-sided limits at $f(0,0)$ are equal: $$\lim_{x\to 0^+} \lim_{y\to 0^+} \frac{f(x,y)}{xy} \stackrel{?}{=} \lim_{y\to 0^+} \lim_{x\to 0^+} \frac{f(x,y)}{xy}$$ (We can assume that both limits exist.)

This can be recognized as the exchange of "one-sided" partial derivatives. However, everything I have found (such as Rudin Theorem 9.41, etc.) considers $f$ defined on an open set, where regular (not one-sided) derivatives can be defined.

Is there a set of simple condition for the above equality to hold? Is there a reference for this?

(My thought right now is to "extend" $f$ to an open set $E \supset [0,1]\times[0,1]$ in a way that preserves the derivatives.)

[Cross-posted from math.stackexchange here]

Consider some $f: [0,1)\times [0,1)\to \mathbb{R}$. I'm interested in conditions that guarantee that the following one-sided second partial derivatives at $(x,y)=(0,0)$ are symmetric: $$ \partial_x^+ \partial_y^+ f(x,y)= \partial_y^+ \partial_x^+ f(x,y). $$ (where as usual $\partial_x^+$ and $\partial_y^+$ indicate partial derivatives defined via one-sided limits).

Everything on the topic that I've found (such as Rudin Theorem 9.41, etc.) considers $f$ defined on an open set, where regular (not one-sided) derivatives can be defined.

Is there a set of simple condition for the above equality to hold? Is there a reference for this?

(My thought right now is to "extend" $f$ to an open set $E \supset [0,1]\times[0,1]$ in a way that preserves the derivatives.)

[Cross-posted from math.stackexchange here]